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% File Name as ATO: Symmetric kNoid

\cl{\bf  Symmetric And Skew-symmetric k-Noids} 
\LF
The symmetric and skew k-noids are parametrized by
k-punctured spheres. The number $k$ of ends is selected
with the 3DXM parameter $ee$.
We use parameter lines that extend polar coordinates
around the punctures. Formulas are taken from [K2]. These formulas
easily allow to change the relative size of neighboring catenoid
ends for the {\bf symmetric 2k-noids}, try the {\it default morph}
which varies the 3DXM parameter $aa$ in $[0,1]$.
The symmetric k-noids, where the number $k$ of ends is odd, 
cannot be deformed in 3DXM.
\lf
The ends of the {\bf skew-symmetric k-noids} in 3DXM are all of the same
size. The {\it default morph} varies the angle between neighboring
ends, again by changing the 3DXM parameter $aa$ in $[0,1]$. 
\lf
The size of the ends is always controlled  using $bb$.
\Lf
  The intersection of the two families are the symmetric k-noids with all
  ends of the same size, a family found by Jorge and Meeks.  
  Their k-noids are the first finite total curvature immersions 
for which the Weierstrass data
were manufactured to fit a previously conceived qualitative
global picture of the surfaces. 
\Lf
The default morph of the symmetric 2k-noids ends at symmetric k-noids
with $k$ punctures as $aa$ goes to~zero. 
%\goodbreak \vfil \eject \ni%
\lf
In the default morph of the skew-symmetric k-noids
the angle between neighboring pairs of ends goes to zero.
Near the limit the 4-noid looks like a pair of parallel catenoids that are joined by
a very thin ``caten\-oidal'' handle. In fact the picture looked so suggestive
that it convinced David Hoffman immediately
that the idea of adding handles might be promising in a wide range of
situations. {\it Try this default morph!}
\Lf
[K2]   H. Karcher, Construction of minimal surfaces, in ``Surveys in
        Geometry'', Univ. of Tokyo, 1989,  and Lecture Notes No.12,
        SFB 256, Bonn, 1989, pp. 1-96.
\Lf
  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].
\Lf
[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993
\Lf
[DHKW] U. Dierkes, S. Hildebrand, A. K\"uster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991
\LF
H. K.

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